arXiv subjects

Guang-Hui Zheng

Publications and source records attributed to Guang-Hui Zheng.

At least 19 recordsLinked to original sources

arXiv2026

Inverse Scattering of Periodic Surfaces with Multilayer Superlenses: Perturbation Analysis and Resolution Enhancement

This paper presents a rigorous mathematical analysis of inverse scattering for periodic surfaces through a multilayer superlens. Using the transformed field expansion, we derive a highly explicit reconstruction formula. We theoretically prove that under perfect matching, the reconstruction factor is independent of the number of layers and depends solely on the total material thicknesses, achieving uniform boundedness when the slab thickness dominates. By establishing first-order perturbation formulas and high-frequency asymptotics, we quantify how small material losses induce exponential amplification of high-frequency errors. Crucially, we show that increasing the number of layers substantially reduces this amplification coefficient. Consequently, for fixed total thicknesses, multilayer structures offer significantly improved stability and extended resolution limits for subwavelength reconstruction.

math.AP↗

arXiv2026

Quantitative Analysis of Second-Harmonic Generation in Plasmonic Nanowires: Symmetry Breaking and a Topological Defect

This paper develops a quantitative theory of second-harmonic generation (SHG) in hollow plasmonic nanowires, distinguishing geometric symmetry breaking from cavity effects. We consider a centrosymmetric material with a concentric circular cavity and a slightly perturbed outer boundary. Relative to the solid nanowire in~\cite{liu2026quantitative}, the cavity introduces a topological defect by changing the cross-section from a disk to a doubly connected domain, whereas the outer-boundary perturbation breaks geometric symmetry. Local inversion-symmetry breaking at the interfaces induces nonlinear surface polarization. Using layer potentials and boundary-perturbation asymptotics, we derive explicit leading- and first-order formulas for the fundamental and second-harmonic fields. For a circular outer boundary, the leading-order second-harmonic field is purely quadrupolar; its amplitude differs from the solid-wire case by $O(γ^2)$ as $γ=r_i/r_e\to0$ and decays as $O(r_e-r_i)$ in the nonresonant thin-shell limit. For perturbations proportional to $\cos(nθ)$, $n\ge3$, the first-order second-harmonic correction contains only modes $n-2$, $n$, and $n+2$, yielding dipolar, quadrupolar, or higher-order radiation for $n=3$, $n=4$, or $n\ge5$, respectively, provided the lowest-order modal coefficient is nonzero. We derive plasmonic resonance conditions at both frequencies and estimate the singular orders of single and simultaneous resonances, including possible higher-order singularities from coincident resonance factors. Thus geometric symmetry determines the multipolar modes, whereas the cavity radius controls their amplitudes and resonant enhancement.

math.AP↗

arXiv2026

Bayesian Geometric Inversion for Wave Equation: Well-Posedness and Multiscale Multimodal Posterior Approximation

Geometric full-waveform inversion aims to recover sharp subsurface interfaces from noisy waveform observations. We formulate this problem within a Bayesian level-set framework, where the unknown interface is represented as zero level set of a latent function endowed with a Gaussian prior. Although the level-set-to-medium-parameter map is discontinuous, we establish that the level-set-to-observation map is continuous almost surely with respect to the prior, thereby guaranteeing that the posterior measure is well-posed and locally Lipschitz continuous with respect to the data in the Hellinger metric. To enable tractable posterior computation under the severe nonlinearity of the inverse problem and the resulting multimodality, we develop a multiscale multimodal Laplace approximation. A frequency-continuation strategy incorporates waveform data progressively from low to high frequencies, identifying and propagating multiple candidate modes across successive scales. At the final frequency level, local Gaussian approximations are constructed around the retained modes and combined into a Gaussian mixture, whose weights are determined by least-squares fitting to the unnormalized posterior density. Numerical experiments demonstrate that the method accurately reconstructs anomalous interfaces while effectively capturing posterior multimodality and the local uncertainty associated with distinct plausible geometries.

math.NA↗

arXiv2026

Geometric Optics Approximation Sampling: A Reflector-Induced Transport Map Framework

In this paper, we propose Geometric Optics Approximation Sampling (GOAS), a reflector-induced transport-map framework for sampling from target measures. Once a reflecting surface is constructed, the associated transport map is explicitly determined by the physical law of reflection. As a concrete realization, we develop a supporting-hyperellipsoid construction that requires only a discrete approximation of the target measure and does not require gradient information of the target density. The formulation accommodates both density-based and sample-based target representations. A softmin smoothing technique is introduced to obtain a smooth approximate transport map from this piecewise hyperellipsoidal construction. We establish well-posedness and stability of the reflector-induced push-forward measure and derive quantitative error estimates in the maximum mean discrepancy metric, and convergence of continuous statistical observables, including fixed-order moments. Numerical experiments on an analytically tractable example, strongly non-Gaussian targets, sample-based target approximations, and Bayesian inverse problems demonstrate the accuracy and flexibility of GOAS.

math.NA↗

arXiv2026

Geometric Optics Approximation Sampling: A Far-Field Reflector-Induced Transport Framework

We develop a far-field geometric optics approximation sampling (GOAS) framework for constructing direct samplers from target measures. The method exploits the connection between the far-field reflector problem and optimal transport with logarithmic cost, leading to a natural primal--dual transport structure. The associated dual reflector provides a reciprocal backward transport and, in the invertible smooth setting, the inverse of the forward reflector map. For numerical realization, we adopt a supporting hyperparaboloid construction based on a discrete approximation of the target measure. This construction is gradient-free with respect to the target density and naturally accommodates both density-based and sample-based target representations. The resulting piecewise reflector admits two sampling realizations: a primal--dual consistency resampling strategy that operates directly on the nonsmooth reflector, and a softmin-regularized realization yielding an explicit smooth transport map through the physical law of reflection. We establish the well-posedness and stability of the reflector-induced sampling measure and derive Wasserstein error estimates. Numerical experiments on non-Gaussian targets and Bayesian inverse problems demonstrate the accuracy, stability, Wasserstein error behavior, and applicability of the proposed framework, and compare it with MCMC and polynomial transport-map methods.

math.NA↗

arXiv2026

Determining internal topological structures and running cost of mean field games with partial boundary measurement

This paper investigates the simultaneous reconstruction of the running cost function and the internal topological structure within the mean-field games (MFG) system utilizing partial boundary data. The inverse problem is notably challenging due to factors such as nonlinear coupling, the necessity for multi-parameter reconstruction, constraints on probability measures, and the limited availability of measurement information. To address these challenges, we propose an innovative approach grounded in a higher-order linearization method. This method is tailored for inverse problems in MFG systems that involve Dirichlet and Neumann boundary conditions. Initially, we present unique reconstruction results for the cost function and internal topological structure of the MFG system under various homogeneous boundary conditions. Subsequently, we extend these results to accommodate inhomogeneous boundary conditions. These findings greatly enhance our understanding of simultaneous reconstruction in complex MFG systems.

math.OC↗

arXiv2026

Vanishing Clutter: Fast and Accurate Shape Imaging via Active Cloaking

Imaging a target sample in near-field scanning optical microscopy (NSOM) is fundamentally limited by measurement artifacts and data contamination from multiple probe-sample scattering. The probe, essential for subwavelength resolution, inherently perturbs the local field, degrading the signal-to-noise ratio and rendering the inverse problem for quantitative shape reconstruction highly ill-posed. We first establish the well-posedness of the corresponding forward model, providing a rigorous foundation for subsequent imaging. We then repurpose active cloaking--conventionally the antagonist of imaging--as an enabling mechanism to eliminate probe-induced interference. Rather than directly reconstructing the sample from corrupted data, we actively cloak the probe by formulating an optimal control problem and prove the existence and stability of its minimizers. Leveraging the theory of localized anomalous resonance in layered plasmonic structures, we derive an exact closed-form minimizer, thereby circumventing the computationally prohibitive iterative solution of the optimal control problem. The resulting cloaking-driven interference removal yields a virtually probe-free measurement environment, enabling fast, artifact-free shape reconstruction. Extensive numerical experiments demonstrate accurate shape reconstruction and dramatic acceleration--often by orders of magnitude--over conventional iterative methods applied directly to probe--contaminated data without cloaking--based preprocessing, validating the robustness and transformative potential of the proposed approach for high-fidelity subwavelength imaging.

math.NA↗

arXiv2026

Quantitative symmetry-breaking and nonlinear harmonic generation in plasmonics

We develop a quantitative mathematical theory that offers new perspectives on nonlinear harmonic generation in plasmonic structures arising from symmetry breaking. Focusing on second harmonic generation--the most fundamental process and the most extensively studied owing to its practical significance--we establish a theoretical framework that can be readily extended to higher-order harmonics. We investigate the plasmonic system in the static regime using a columnar nanowire with $n$-fold rotational symmetry ($n \in \mathbb{N}$) and construct a phenomenological model in which the second harmonic response originates from nonlinear sources confined to a selvedge region near the surface. By introducing a notion of symmetry degree grounded in group theory, we precisely quantify the second harmonic generation in terms of multipolar contributions. Our theory complements existing physical descriptions of this practically important phenomenon and provides a rigorous account of how nonlinear optical efficiency depends on shape, size, symmetry, and defects in plasmonic structures.

math.AP↗

arXiv2026

Multi-parameter determination in the semilinear Helmholtz equation

This paper studies an inverse boundary value problem for a semilinear Helmholtz equation with Neumann boundary conditions in a bounded domain $Ω\subset \mathbb{R}^n$ ($n\ge2$). The objective is to recover the unknown linear and nonlinear coefficients from the associated Neumann-to-Dirichlet (NtD) map. Using a higher-order linearization approach, we establish the unique determination of both coefficients from boundary measurements. For spatial dimensions $n\ge3$, uniqueness holds under $C^γ(\overlineΩ)$ regularity assumptions with $0<γ<1$, while in the two-dimensional case uniqueness is obtained under Sobolev regularity $W^{1,p}(Ω)$ with $p>2$. The analysis relies on the well-posedness of the forward problem together with techniques from linear inverse problems, including Runge-type approximation arguments and Fourier analysis. In addition, we develop a numerical reconstruction framework for recovering the coefficients from boundary data. The forward problem is discretized using a finite difference scheme combined with a quasi-Newton iteration, and the inverse problem is formulated within a Bayesian inference framework. Posterior distributions of the coefficients are explored using the preconditioned Crank-Nicolson (pCN) Markov chain Monte Carlo algorithm, which provides both point estimates and uncertainty quantification. Numerical experiments demonstrate the effectiveness of the proposed reconstruction method and illustrate the theoretical uniqueness results.

math.NA↗

arXiv2025

On surface polariton resonance and its curvature concentration effects from 3D elastic nanorods

This paper investigates surface polariton resonance (SPR) in three-dimensional elastic metamaterials with nanorod geometry. The primary motivation is to surpass the physical limitations imposed by the quasi-static approximation for SPRs through anisotropic geometric design. The analysis boils down to analyzing the spectral properties of the matrix-valued elastic Neumann-Poincaré (NP) operator defined on the nanorod boundary. We develop novel analytical techniques and conduct a rigorous asymptotic analysis of elastic layer potential operators, specifically adapted for highly anisotropic structures. Within this framework, we derive precise asymptotic formulas for the scattered field in the quasi-static regime. A thorough examination of these expressions yields explicit resonance conditions that intricately link three fundamental parameters: elastic material parameters, wave frequency, and nanorod geometry. Furthermore, we characterize the intrinsic relationship between these parameters and the associated energy blow-up rate of the resonant field. This analysis explicitly establishes a sharp curvature concentration effect at the nanorod extremities, where field enhancement is locally maximized. Our work provides a rigorous theoretical foundation for harnessing elastic SPRs through anisotropic geometric engineering, with implications for sensing, wave focusing, and metamaterial applications.

math-ph↗

arXiv2025

Inverse Random Source Problem for the Helmholtz Equation from Statistical Phaseless Data

This paper investigates the problem of reconstructing a random source from statistical phaseless data for the two-dimensional Helmholtz equation. The major challenge of this problem is non-uniqueness, which we overcome through a reference source technique. Firstly, we introduce some artificially added point sources into the inverse random source system and derive phase retrieval (PR) formulas for the expectation and variance of the radiated fields. This paper rigorously analyze the uniqueness and stability of the recovered statistics of the radiated fields. Afterwards, since the direct problem has a unique mild solution, by examining the expectation and variance of this solution and combined with the phase retrieval formulas, we derive the Fredholm integral equations to solve the inverse random source problem (IRSP). We prove the stability of the corresponding integral equations. To quantify the uncertainty of the random source, we utilize the Bayesian method to reconstruct the random source and establish the well-posedness of the posterior distribution. Finally, numerical experiments demonstrate the effectiveness of the proposed method and validate the theoretical results.

math.NA↗

arXiv2025

Microscale Hydrodynamic Cloaking via Geometry Design in a Depth-Varying Hele-Shaw Cell

We theoretically and numerically demonstrate that hydrodynamic cloaking can be achieved by simply adjusting the geometric depth of a region surrounding an object in microscale flow, rendering the external flow field undisturbed. Using the depth-averaged model, we develop a theoretical framework based on analytical solutions for circular and confocal elliptical cloaks. For cloaks of arbitrary shape, we employ an optimization method to determine the optimal depth profile within the cloaking region. Furthermore, we propose a multi-object hydrodynamic cloak design incorporating neutral inclusion theory. All findings are validated numerically. The presented cloaks feature simpler structures than their metamaterial-based counterparts and offer straightforward fabrication, thus holding significant potential for microfluidic applications.

physics.flu-dyn↗

arXiv2025

Hybridization theory for plasmon resonance in metallic nanostructures

In this paper, we investigate the hybridization theory of plasmon resonance in metallic nanostructures, which has been validated by the authors in [31] through a series of experiments. In an electrostatic field, we establish a mathematical framework for the Neumann-Poincaré(NP) type operators for metallic nanoparticles with general geometries related to core and shell scales. We calculate the plasmon resonance frequency of concentric disk metal nanoshells with normal perturbations at the interfaces by the asymptotic analysis and perturbation theory to reveal the intrinsic hybridization between solid and cavity plasmon modes. The theoretical finding are convincingly supported by extensive numerical experiments. Our theory corroborates and strengthens that by properly enriching the materials structures as well as the underlying geometries, one can induce much richer plasmon resonance phenomena of practical significance.

math.AP↗

arXiv2025

Solving Bayesian inverse problems via Fisher adaptive Metropolis adjusted Langevin algorithm

The preconditioned Metropolis adjusted Langevin algorithm (MALA) is a widely used method in statistical applications, where the choice of the preconditioning matrix plays a critical role. Recently, Titsias \cite{Titsias2024} demonstrated that the inverse Fisher information matrix is the optimal preconditioner by minimizing the expected squared jump distance and proposed an adaptive scheme to estimate the Fisher matrix using the sampling history. In this paper, we apply the Fisher adaptive Metropolis adjusted Langevin algorithm (MALA) to Bayesian inverse problems. Moreover, we provide a rigorous convergence rate analysis for the adaptive scheme used to estimate the Fisher matrix. To evaluate its performance, we use this algorithm to sample from posterior distributions in several Bayesian inverse problems. And compare its constructions with the standard adaptive Metropolis adjusted Langevin algorithm (which employs the empirical covariance matrix of the posterior distribution as the preconditioner) and the preconditioned Crank-Nicolson (pCN) algorithm. Our numerical results demonstrate show that the Fisher adaptive MALA is highly effective for Bayesian inversion, and significantly outperforms other sampling methods, particularly in high-dimensional settings.

math.NA↗

arXiv2024

Determination and reconstruction of a semilinear term from point measurements

In this article we study the inverse problem of determining a semilinear term appearing in an elliptic equation from boundary measurements. Our main objective is to develop flexible and general theoretical results that can be used for developing numerical reconstruction algorithm for this inverse problem. For this purpose, we develop a new method, based on different properties of solutions of elliptic equations, for treating the determination of the semilinear term as a source term from a point measurement of the solutions. This approach not only allows us to make important relaxations on the data used so far for solving this class of inverse problems, including general Dirichlet excitation lying in a space of dimension one and measurements located at one point on the boundary of the domain, but it also allows us to derive a novel algorithm for the reconstruction of the semilinear term. The effectiveness of our algorithm is corroborated by extensive numerical experiments. Notably, as demonstrated by the theoretical analysis, we are able to effectively reconstruct the unknown nonlinear source term by utilizing solely the information provided by the measurement data at a single point.

math.NA↗

arXiv2024

Phaseless uniqueness for determining internal source in photo-thermal effect

The paper investigates an inverse problem of recovering the internal source from external temperature measurements in photo-thermal effect. The photo-thermal effect actually involves two physical processes: electromagnetic scattering and heat transfer, described by a nonlinear coupled system of Maxwell's equation and the heat transfer equation. The nonlinear coupling term in the system is represented by the square of the modulus of the electromagnetic (missing the phase information of the electromagnetic field), and the absence of this phase information poses a significant challenge to the reconstruction of the internal source. In addition, the interaction and mutual influence of multiple physical fields, including electric field, magnetic field and temperature field, add to the complexity involved in the inversion of the internal source. Based on the potential theory and asymptotic analysis, we prove that the internal source can be uniquely determined up to sign by the external temperature field. This provides a solid theoretical basis for designing the internal source inversion algorithm and further exploring the theoretical aspects of photo-thermal effect.

math.AP↗

arXiv2024

Simultaneously Cloaking Electric and Hydrodynamic Fields via Electro-osmosis

In this paper, we develop a general mathematical framework for the electro-osmosis problem to design simultaneous microscale electric and hydrodynamic cloaking in a Hele-Shaw configuration. A novel approach to achieving simultaneously cloaking both the electric and flow fields through a combination of scattering-cancellation technology and an electro-osmosis effect is proposed. In the design, the electric field is manipulated with scattering-cancellation technology while the pressure with electro-osmosis effect. As proof of this concept, the perfect electric and hydrodynamic cloaking conditions are derived for the cloaks with the cross-sectional shape being annulus or confocal ellipses using the layer potential techniques. Furthermore, we also propose an optimization scheme for the design of approximate cloaks within general geometries and prove the well-posedness of the optimization problem. In particular, the conditions that can ensure the simultaneous occurrence of approximate cloaks for general geometries are also established. Our theoretical findings are validated by a variety of numerical results and guide efficiently designing electric-related multiphysics cloaking.

math.AP↗

arXiv2023

Enhanced Microscale Hydrodynamic Near-cloaking using Electro-osmosis

In this paper, we develop a general mathematical framework for enhanced hydrodynamic near-cloaking of electro-osmotic flow for more complex shapes, which is obtained by simultaneously perturbing the inner and outer boundaries of the perfect cloaking structure. We first derive the asymptotic expansions of perturbed fields and obtain a first-order coupled system. We then establish the representation formula of the solution to the first-order coupled system using the layer potential techniques. Based on the asymptotic analysis, the enhanced hydrodynamic near-cloaking conditions are derived for the control region with general cross-sectional shape. The conditions reveal the inner relationship between the shapes of the object and the control region. Especially, for the shape of a deformed annulus or confocal ellipses cylinder, the cloaking conditions and relationship of shapes are quantified more accurately. Our theoretical findings are validated and supplemented by a variety of numerical results. The results in this paper also provide a mathematical foundation for more complex hydrodynamic cloaking.

math.AP↗